New applications of random sampling in computational geometry
TL;DR: This paper gives several new demonstrations of the usefulness of random sampling techniques in computational geometry by creating a search structure for arrangements of hyperplanes by sampling the hyperplanes and using information from the resulting arrangement to divide and conquer.
Abstract: This paper gives several new demonstrations of the usefulness of random sampling techniques in computational geometry. One new algorithm creates a search structure for arrangements of hyperplanes by sampling the hyperplanes and using information from the resulting arrangement to divide and conquer. This algorithm requiresO(sd+?) expected preprocessing time to build a search structure for an arrangement ofs hyperplanes ind dimensions. The expectation, as with all expected times reported here, is with respect to the random behavior of the algorithm, and holds for any input. Given the data structure, and a query pointp, the cell of the arrangement containingp can be found inO(logs) worst-case time. (The bound holds for any fixed ?>0, with the constant factors dependent ond and ?.) Using point-plane duality, the algorithm may be used for answering halfspace range queries. Another algorithm finds random samples of simplices to determine the separation distance of two polytopes. The algorithm uses expectedO(n[d/2]) time, wheren is the total number of vertices of the two polytopes. This matches previous results [10] for the cased = 3 and extends them. Another algorithm samples points in the plane to determine their orderk Voronoi diagram, and requires expectedO(s1+?k) time fors points. (It is assumed that no four of the points are cocircular.) This sharpens the boundO(sk2 logs) for Lee's algorithm [21], andO(s2 logs+k(s?k) log2s) for Chazelle and Edelsbrunner's algorithm [4]. Finally, random sampling is used to show that any set ofs points inE3 hasO(sk2 log8s/(log logs)6) distinctj-sets withj≤k. (ForS ?Ed, a setS? ?S with |S?| =j is aj-set ofS if there is a half-spaceh+ withS? =S ?h+.) This sharpens with respect tok the previous boundO(sk5) [5]. The proof of the bound given here is an instance of a "probabilistic method" [15].
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TL;DR: The expectation of the k th degree average of the number of lines intersecting a triangle is O(n/r) for any fixed k, which is the constant of proportionality in this result.
Abstract: A theorem of Chazelle and Friedman with numerous applications in combinatorial and computational geometry asserts that for any set L of n lines in the plane and for any parameter r>1 there exists a subdivision of the plane into at most Cr2 (possibly unbounded) triangles, C a constant, such that the interior of each triangle is intersected by at most n/r lines of L . (Such a subdivision is called a (1/r) -cutting for L .) We give upper and lower bounds on the constant C . We also consider the canonical triangulation of the arrangement of a random sample of r lines from L . Although this typically is not a (1/r) -cutting, the expectation of the k th degree average of the number of lines intersecting a triangle is O(n/r) for any fixed k . We estimate the constant of proportionality in this result.
24 citations
Cites methods from "New applications of random sampling..."
...This (asymptotically optimal) result was first proved by Chazelle and Friedman [8] by a probabilistic method, improving a previous slightly weaker bound of O.r 2 log 2 r/ proved by Clarkson [ 9 ] and implicitly contained also in the paper Haussler and Welzl [16]....
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01 Aug 1997
TL;DR: This work describes the first known algorithm for efficiently maintaining a Binary Space Partition (BSP) for n continuously moving segments in the plane, whose interiors remain disjoint throughout the motion.
Abstract: We describe the first known algorithm for efficiently maintaining a Binary Space Partition (BSP) for n continuously moving segments in the plane, whose interiors remain disjoint throughout the motion. Under reasonable assumptions on the motion, we show that the total number of times this BSP changes is O.n 2 / ,a nd that we can update the BSP in O.logn/ expected time per change. Throughout the motion, the expected size of the BSP is O.n logn/. We also consider the problem of constructing a BSP for n static triangles with pairwise-disjoint interiors in R 3 . We present a randomized algorithm that constructs a BSP of size O.n 2 / in O.n 2 log 2 n/ expected time. We also describe a deterministic algorithm that constructs a BSP of size O..nCk/log 2 n/ and height O.logn/ in O..nCk/log 3 n/ time, where k is the number of intersection points between the edges of the projections of the triangles onto the xy-plane. This is the first known algorithm that constructs a BSP of O.logn/ height for disjoint triangles inR 3 . © 2000 Elsevier Science B.V. All rights reserved.
24 citations
TL;DR: The δ-relativeε-approximation method, developed for the CRCW variant of the PRAM parallel computation model, can be easily implemented to run in $O(\log n(\log\log n)^{d-1})$ time using linear work on an EREW PRAM.
Abstract: We give fast and efficient methods for constructing e-nets and e-approximations for range spaces with bounded VC-exponent. These combinatorial structures have wide applicability to geometric partitioning problems, which are often used in divide-and-conquer constructions in computational geometry algorithms. In addition, we introduce a new deterministic set approximation for range spaces with bounded VC-exponent, which we call the δ-relativee-approximation, and we show how such approximations can be efficiently constructed in parallel. To demonstrate the utility of these constructions we show how they can be used to solve the linear programming problem in \({\Bbb R}^d\) deterministically in \(O((\log\log n)^d)\) time using linear work in the PRAM model of computation, for any fixed constant d. Our method is developed for the CRCW variant of the PRAM parallel computation model, and can be easily implemented to run in \(O(\log n(\log\log n)^{d-1})\) time using linear work on an EREW PRAM.
24 citations
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TL;DR: In this article, a simple randomized algorithm for constructing the convex hull of a set of n points in the plane with expected running time O(nlogh) where h is the number of points on the hull.
Abstract: This paper contains a simple, randomized algorithm for constructing the convex hull of a set ofn points in the plane with expected running timeO(nlogh) whereh is the number of points on the convex hull.
24 citations
01 Jul 1992
TL;DR: A new technique for geometric enumeration problems is proposed which iteratively reduces the search space by a half and provides efficient algorithms.
Abstract: We study the problem of enumerating k farthest pairs for n points in the plane and the problems of enumerating k closest/farthest bichromatic pairs of n red and n blue points in the plane. We propose a new technique for geometric enumeration problems which iteratively reduces the search space by a half and provides efficient algorithms. As applications of this technique, we develop algorithms, using higher order Voronoi diagrams, for the above problems, which run in O(min{n2, n log n + k4/3 log n/log1/3k}) time and O(n+k4/3/(log k)1/3+k log n) space. Since, to the authors' knowledge, no nontrivial algorithms have been known for these problems, our algorithms are currently fastest when k=o(n3/2).
23 citations
References
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Book•
01 Jan 1968
TL;DR: The arrangement of this invention provides a strong vibration free hold-down mechanism while avoiding a large pressure drop to the flow of coolant fluid.
Abstract: A fuel pin hold-down and spacing apparatus for use in nuclear reactors is disclosed. Fuel pins forming a hexagonal array are spaced apart from each other and held-down at their lower end, securely attached at two places along their length to one of a plurality of vertically disposed parallel plates arranged in horizontally spaced rows. These plates are in turn spaced apart from each other and held together by a combination of spacing and fastening means. The arrangement of this invention provides a strong vibration free hold-down mechanism while avoiding a large pressure drop to the flow of coolant fluid. This apparatus is particularly useful in connection with liquid cooled reactors such as liquid metal cooled fast breeder reactors.
17,939 citations
01 Jan 1985
TL;DR: This book offers a coherent treatment, at the graduate textbook level, of the field that has come to be known in the last decade or so as computational geometry.
Abstract: From the reviews: "This book offers a coherent treatment, at the graduate textbook level, of the field that has come to be known in the last decade or so as computational geometry...The book is well organized and lucidly written; a timely contribution by two founders of the field. It clearly demonstrates that computational geometry in the plane is now a fairly well-understood branch of computer science and mathematics. It also points the way to the solution of the more challenging problems in dimensions higher than two."
6,525 citations
"New applications of random sampling..." refers background in this paper
...(In fact the mapping γ is not unique in this regard: see [13, 23, 2]....
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Book•
01 Jan 1973
4,243 citations
TL;DR: This chapter reproduces the English translation by B. Seckler of the paper by Vapnik and Chervonenkis in which they gave proofs for the innovative results they had obtained in a draft form in July 1966 and announced in 1968 in their note in Soviet Mathematics Doklady.
Abstract: This chapter reproduces the English translation by B. Seckler of the paper by Vapnik and Chervonenkis in which they gave proofs for the innovative results they had obtained in a draft form in July 1966 and announced in 1968 in their note in Soviet Mathematics Doklady. The paper was first published in Russian as Вапник В. Н. and Червоненкис А. Я. О равномерноЙ сходимости частот появления событиЙ к их вероятностям. Теория вероятностеЙ и ее применения 16(2), 264–279 (1971).
3,939 citations
"New applications of random sampling..." refers background in this paper
...Vapnik and Chervonenkis [27] have derived general conditions under which several probabilities may be uniformly estimated using one random sample....
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Book•
01 Jan 1978TL;DR: In this article, the authors present a coherent treatment of computational geometry in the plane, at the graduate textbook level, and point out the way to the solution of the more challenging problems in dimensions higher than two.
Abstract: From the reviews: "This book offers a coherent treatment, at the graduate textbook level, of the field that has come to be known in the last decade or so as computational geometry...The book is well organized and lucidly written; a timely contribution by two founders of the field. It clearly demonstrates that computational geometry in the plane is now a fairly well-understood branch of computer science and mathematics. It also points the way to the solution of the more challenging problems in dimensions higher than two."
3,419 citations